Let \(P_w(n)\) be the set of distinct subwords (intervals) in a word \(w\).
Let \(p_w(n)\) be the cardinality of \(P_w(n)\).
Let \(f(c)\) be the sequence in FASTA with 4-symbol Protein Data Bank code \(c\).
\(|P_{f(3WHL_1)}(2) \setminus P_{f(4MQA_1)}(2)|=76\),
\(|P_{f(4MQA_1)}(2) \setminus P_{f(3WHL_1)}(2)|=89\).
Let
\(
Z_k(x,y)=|P_x(k)\setminus P_y(k)|+|P_y(k)\setminus P_x(k)|
\)
be a LZ76 style (set of subwords) Jaccard distance numerator for \(P(k)\).Hydrophobic-polar version of Sequence 1:100111011001010000111100010010011011100101100111011011110111101001110111001010110111001100011011011001101100011011110010111100100001100010001101110101101010101111000101101111011010011011110010010110100001000001010011000001011010110101111110010001000011011001010000010101
Pair
\(Z_2\)
Length of longest common subsequence
3WHL_1,4MQA_1
165
3
3WHL_1,1WXS_1
152
4
4MQA_1,1WXS_1
179
3
Newick tree
[
4MQA_1:89.17,
[
3WHL_1:76,1WXS_1:76
]:13.17
]
Let d be the
Otu--Sayood
distance d.
Let d1 be the Otu--Sayood distance d1. (This makes the 4TYN sequence AAAAAA a close match...)
A roughly speaking expected distance is \((0.85)(0.8)(\frac{515
}{\log_{20}
515}-\frac{245}{\log_{20}245})=77.2\)
Status
Protein1
Protein2
d
d1/2
Query variables
3WHL_1
4MQA_1
96
93
Was not able to put for d Was not able to put for d1
In notation analogous to
[Theorem 16, Kjos-Hanssen, Niraula and Yoon (2022)],
\[
\delta=
\alpha \mathrm{min} + (1-\alpha) \mathrm{max}=
\begin{cases}
d &\alpha=0,\\
d_1/2 &\alpha=1/2
\end{cases}
\]